# Classical modular forms downloaded from the LMFDB on 17 October 2026. # Search link: https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/1860/ # Query "{'level': 1860}" returned 64 forms, sorted by analytic conductor. # Each entry in the following data list has the form: # [Label, Dim, $A$, Field, CM, RM, Traces, Fricke sign, $q$-expansion] # For more details, see the definitions at the bottom of the file. "1860.1.l.a" 1 0.9282609234973844 "1.1.1.1" [-15, -1860] [124] [-1, -1, 1, 0] NULL "q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}-q^{8}+\\cdots" "1860.1.l.b" 1 0.9282609234973844 "1.1.1.1" [-15, -1860] [124] [-1, 1, 1, 0] NULL "q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}-q^{8}+\\cdots" "1860.1.l.c" 1 0.9282609234973844 "1.1.1.1" [-15, -1860] [124] [1, -1, -1, 0] NULL "q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}+q^{8}+\\cdots" "1860.1.l.d" 1 0.9282609234973844 "1.1.1.1" [-15, -1860] [124] [1, 1, -1, 0] NULL "q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}+q^{8}+\\cdots" "1860.1.be.a" 2 0.9282609234973844 "2.0.3.1" [-15] [] [-1, -1, 1, 0] NULL "q+\\zeta_{6}^{2}q^{2}+\\zeta_{6}^{2}q^{3}-\\zeta_{6}q^{4}+\\zeta_{6}q^{5}+\\cdots" "1860.1.be.b" 2 0.9282609234973844 "2.0.3.1" [-15] [] [-1, 1, 1, 0] NULL "q-\\zeta_{6}q^{2}-\\zeta_{6}^{2}q^{3}+\\zeta_{6}^{2}q^{4}+\\zeta_{6}q^{5}+\\cdots" "1860.1.be.c" 2 0.9282609234973844 "2.0.3.1" [-15] [] [1, -1, -1, 0] NULL "q+\\zeta_{6}q^{2}+\\zeta_{6}^{2}q^{3}+\\zeta_{6}^{2}q^{4}-\\zeta_{6}q^{5}+\\cdots" "1860.1.be.d" 2 0.9282609234973844 "2.0.3.1" [-15] [] [1, 1, -1, 0] NULL "q-\\zeta_{6}^{2}q^{2}-\\zeta_{6}^{2}q^{3}-\\zeta_{6}q^{4}-\\zeta_{6}q^{5}+\\cdots" "1860.1.bt.a" 4 0.9282609234973844 "4.0.125.1" [-15] [] [-1, -1, -4, 0] NULL "q+\\zeta_{10}^{2}q^{2}-\\zeta_{10}q^{3}+\\zeta_{10}^{4}q^{4}-q^{5}+\\cdots" "1860.1.bt.b" 4 0.9282609234973844 "4.0.125.1" [-15] [] [-1, 1, -4, 0] NULL "q-\\zeta_{10}q^{2}+\\zeta_{10}q^{3}+\\zeta_{10}^{2}q^{4}-q^{5}+\\cdots" "1860.1.bt.c" 4 0.9282609234973844 "4.0.125.1" [-15] [] [1, -1, 4, 0] NULL "q+\\zeta_{10}q^{2}-\\zeta_{10}q^{3}+\\zeta_{10}^{2}q^{4}+q^{5}+\\cdots" "1860.1.bt.d" 4 0.9282609234973844 "4.0.125.1" [-15] [] [1, 1, 4, 0] NULL "q-\\zeta_{10}^{2}q^{2}+\\zeta_{10}q^{3}+\\zeta_{10}^{4}q^{4}+q^{5}+\\cdots" "1860.1.df.a" 8 0.9282609234973844 "8.0.1265625.1" [-15] [] [-1, -1, -4, 0] NULL "q-\\zeta_{30}^{8}q^{2}-\\zeta_{30}^{14}q^{3}-\\zeta_{30}q^{4}+\\cdots" "1860.1.df.b" 8 0.9282609234973844 "8.0.1265625.1" [-15] [] [-1, 1, -4, 0] NULL "q-\\zeta_{30}^{4}q^{2}+\\zeta_{30}^{14}q^{3}+\\zeta_{30}^{8}q^{4}+\\cdots" "1860.1.df.c" 8 0.9282609234973844 "8.0.1265625.1" [-15] [] [1, -1, 4, 0] NULL "q+\\zeta_{30}^{4}q^{2}-\\zeta_{30}^{14}q^{3}+\\zeta_{30}^{8}q^{4}+\\cdots" "1860.1.df.d" 8 0.9282609234973844 "8.0.1265625.1" [-15] [] [1, 1, 4, 0] NULL "q+\\zeta_{30}^{8}q^{2}+\\zeta_{30}^{14}q^{3}-\\zeta_{30}q^{4}+\\cdots" "1860.2.a.a" 1 14.85217477595815 "1.1.1.1" [] [] [0, -1, 1, 0] 1 "q-q^{3}+q^{5}+q^{9}-2q^{13}-q^{15}-4q^{17}+\\cdots" "1860.2.a.b" 1 14.85217477595815 "1.1.1.1" [] [] [0, 1, 1, -2] 1 "q+q^{3}+q^{5}-2q^{7}+q^{9}-4q^{11}-4q^{13}+\\cdots" "1860.2.a.c" 1 14.85217477595815 "1.1.1.1" [] [] [0, 1, 1, 2] -1 "q+q^{3}+q^{5}+2q^{7}+q^{9}-4q^{11}+4q^{13}+\\cdots" "1860.2.a.d" 2 14.85217477595815 "2.2.24.1" [] [] [0, -2, -2, 0] 1 "q-q^{3}-q^{5}+\\beta q^{7}+q^{9}+(-2-\\beta )q^{13}+\\cdots" "1860.2.a.e" 2 14.85217477595815 "2.2.12.1" [] [] [0, 2, -2, 2] 1 "q+q^{3}-q^{5}+(1+\\beta )q^{7}+q^{9}-4q^{11}+\\cdots" "1860.2.a.f" 3 14.85217477595815 "3.3.564.1" [] [] [0, -3, -3, 4] -1 "q-q^{3}-q^{5}+(2-\\beta _{1}+\\beta _{2})q^{7}+q^{9}+\\cdots" "1860.2.a.g" 3 14.85217477595815 "3.3.404.1" [] [] [0, 3, -3, -2] -1 "q+q^{3}-q^{5}+(-1+\\beta _{2})q^{7}+q^{9}+2\\beta _{2}q^{11}+\\cdots" "1860.2.a.h" 3 14.85217477595815 "3.3.7636.1" [] [] [0, 3, 3, 0] -1 "q+q^{3}+q^{5}-\\beta _{1}q^{7}+q^{9}+2q^{11}+\\cdots" "1860.2.a.i" 4 14.85217477595815 "4.4.224148.1" [] [] [0, -4, 4, -4] -1 "q-q^{3}+q^{5}+(-1+\\beta _{1})q^{7}+q^{9}+(\\beta _{1}+\\cdots)q^{11}+\\cdots" "1860.2.g.a" 14 14.85217477595815 NULL [] [] [0, 0, -2, 0] NULL "q+\\beta _{8}q^{3}-\\beta _{1}q^{5}+(-\\beta _{7}-\\beta _{8})q^{7}+\\cdots" "1860.2.g.b" 14 14.85217477595815 NULL [] [] [0, 0, -2, 0] NULL "q-\\beta _{1}q^{3}+\\beta _{13}q^{5}+(-\\beta _{2}+\\beta _{6}-\\beta _{9}+\\cdots)q^{7}+\\cdots" "1860.2.i.a" 16 14.85217477595815 NULL [-155] [] [0, 0, 0, 0] NULL "q-\\beta _{10}q^{3}+\\beta _{5}q^{5}+\\beta _{8}q^{9}-\\beta _{12}q^{13}+\\cdots" "1860.2.i.b" 48 14.85217477595815 NULL [] [] [0, 0, 0, 0] NULL NULL "1860.2.o.a" 44 14.85217477595815 NULL [] [] [0, 0, 0, 8] NULL NULL "1860.2.q.a" 2 14.85217477595815 "2.0.3.1" [] [] [0, -1, -1, -4] NULL "q+(-1+\\zeta_{6})q^{3}-\\zeta_{6}q^{5}+(-4+4\\zeta_{6})q^{7}+\\cdots" "1860.2.q.b" 2 14.85217477595815 "2.0.3.1" [] [] [0, -1, -1, 0] NULL "q+(-1+\\zeta_{6})q^{3}-\\zeta_{6}q^{5}-\\zeta_{6}q^{9}+4\\zeta_{6}q^{11}+\\cdots" "1860.2.q.c" 2 14.85217477595815 "2.0.3.1" [] [] [0, -1, -1, 2] NULL "q+(-1+\\zeta_{6})q^{3}-\\zeta_{6}q^{5}+(2-2\\zeta_{6})q^{7}+\\cdots" "1860.2.q.d" 2 14.85217477595815 "2.0.3.1" [] [] [0, 1, -1, -2] NULL "q+(1-\\zeta_{6})q^{3}-\\zeta_{6}q^{5}+(-2+2\\zeta_{6})q^{7}+\\cdots" "1860.2.q.e" 4 14.85217477595815 "4.0.1089.1" [] [] [0, 2, 2, 0] NULL "q+(1-\\beta _{1})q^{3}+\\beta _{1}q^{5}-\\beta _{1}q^{9}+(3\\beta _{1}+\\cdots)q^{11}+\\cdots" "1860.2.q.f" 6 14.85217477595815 "6.0.2958147.4" [] [] [0, -3, -3, 0] NULL "q+\\beta _{3}q^{3}+(-1-\\beta _{3})q^{5}+(\\beta _{1}-\\beta _{2}+\\cdots)q^{7}+\\cdots" "1860.2.q.g" 6 14.85217477595815 "6.0.64616643.2" [] [] [0, 3, 3, -4] NULL "q+(1+\\beta _{1})q^{3}-\\beta _{1}q^{5}+(-1-\\beta _{1}+\\beta _{2}+\\cdots)q^{7}+\\cdots" "1860.2.q.h" 8 14.85217477595815 "8.0.102293147889.2" [] [] [0, 4, -4, 6] NULL "q+\\beta _{4}q^{3}+(-1+\\beta _{4})q^{5}+(-\\beta _{1}+2\\beta _{4}+\\cdots)q^{7}+\\cdots" "1860.2.q.i" 12 14.85217477595815 NULL [] [] [0, -6, 6, -2] NULL "q-\\beta _{7}q^{3}+(1-\\beta _{7})q^{5}+(-\\beta _{4}-\\beta _{9}+\\cdots)q^{7}+\\cdots" "1860.2.s.a" 64 14.85217477595815 NULL [] [] [0, 0, 0, -8] NULL NULL "1860.2.z.a" 4 14.85217477595815 "4.0.125.1" [] [] [0, 1, -4, 4] NULL "q+(1-\\zeta_{10}+\\zeta_{10}^{2}-\\zeta_{10}^{3})q^{3}-q^{5}+\\cdots" "1860.2.z.b" 16 14.85217477595815 NULL [] [] [0, 4, -16, -3] NULL "q+\\beta _{10}q^{3}-q^{5}-\\beta _{1}q^{7}-\\beta _{9}q^{9}+(-\\beta _{1}+\\cdots)q^{11}+\\cdots" "1860.2.z.c" 20 14.85217477595815 NULL [] [] [0, -5, -20, 5] NULL "q+(-1+\\beta _{6}+\\beta _{8}+\\beta _{10})q^{3}-q^{5}+\\cdots" "1860.2.z.d" 20 14.85217477595815 NULL [] [] [0, -5, 20, -5] NULL "q+\\beta _{8}q^{3}+q^{5}+(-\\beta _{1}-\\beta _{3}-\\beta _{11}+\\cdots)q^{7}+\\cdots" "1860.2.z.e" 20 14.85217477595815 NULL [] [] [0, 5, 20, -1] NULL "q-\\beta _{5}q^{3}+q^{5}-\\beta _{15}q^{7}+\\beta _{2}q^{9}+(-\\beta _{3}+\\cdots)q^{11}+\\cdots" "1860.2.bb.a" 4 14.85217477595815 "4.0.144.1" [] [] [0, 0, 0, -4] NULL "q+(-\\zeta_{12}+2\\zeta_{12}^{3})q^{3}+(\\zeta_{12}-\\zeta_{12}^{3})q^{5}+\\cdots" "1860.2.bb.b" 80 14.85217477595815 NULL [] [] [0, 0, 0, 12] NULL NULL "1860.2.bj.a" 64 14.85217477595815 NULL [] [] [0, 0, 0, 0] NULL NULL "1860.3.e.a" 64 50.68132917101522 NULL [] [] [0, 0, 0, 0] NULL NULL "1860.3.k.a" 44 50.68132917101522 NULL [] [] [0, 0, 0, -16] NULL NULL "1860.3.m.a" 80 50.68132917101522 NULL [] [] [0, -8, 0, 0] NULL NULL "1860.3.bf.a" 40 50.68132917101522 NULL [] [] [0, -60, 0, 0] NULL NULL "1860.3.bf.b" 44 50.68132917101522 NULL [] [] [0, 66, 0, 8] NULL NULL "1860.4.a.a" 1 109.74355261067754 "1.1.1.1" [] [] [0, -3, 5, -8] -1 "q-3q^{3}+5q^{5}-8q^{7}+9q^{9}-20q^{11}+\\cdots" "1860.4.a.b" 5 109.74355261067754 NULL [] [] [0, -15, 25, 1] -1 "q-3q^{3}+5q^{5}+(-\\beta _{3}-\\beta _{4})q^{7}+9q^{9}+\\cdots" "1860.4.a.c" 6 109.74355261067754 NULL [] [] [0, 18, 30, -19] -1 "q+3q^{3}+5q^{5}+(-3-\\beta _{2}-\\beta _{4})q^{7}+\\cdots" "1860.4.a.d" 7 109.74355261067754 NULL [] [] [0, -21, -35, 7] -1 "q-3q^{3}-5q^{5}+(1-\\beta _{5})q^{7}+9q^{9}+\\cdots" "1860.4.a.e" 7 109.74355261067754 NULL [] [] [0, 21, -35, 19] -1 "q+3q^{3}-5q^{5}+(3+\\beta _{2})q^{7}+9q^{9}+\\cdots" "1860.4.a.f" 8 109.74355261067754 NULL [] [] [0, -24, -40, 35] 1 "q-3q^{3}-5q^{5}+(4-\\beta _{1})q^{7}+9q^{9}+\\cdots" "1860.4.a.g" 8 109.74355261067754 NULL [] [] [0, 24, -40, -9] 1 "q+3q^{3}-5q^{5}+(-1-\\beta _{1})q^{7}+9q^{9}+\\cdots" "1860.4.a.h" 9 109.74355261067754 NULL [] [] [0, -27, 45, -35] 1 "q-3q^{3}+5q^{5}+(-4+\\beta _{1})q^{7}+9q^{9}+\\cdots" "1860.4.a.i" 9 109.74355261067754 NULL [] [] [0, 27, 45, 9] 1 "q+3q^{3}+5q^{5}+(1-\\beta _{6})q^{7}+9q^{9}+\\cdots" "1860.4.g.a" 46 109.74355261067754 NULL [] [] [0, 0, 20, 0] NULL NULL "1860.4.g.b" 46 109.74355261067754 NULL [] [] [0, 0, 20, 0] NULL NULL # Label -- # The **label** of a newform $f\in S_k^{\rm new}(N,\chi)$ has the format \( N.k.a.x \), where # - \( N\) is the level; # - \(k\) is the weight; # - \(N.a\) is the label of the Galois orbit of the Dirichlet character $\chi$; # - \(x\) is the label of the Galois orbit of the newform $f$. # For each embedding of the coefficient field of $f$ into the complex numbers, the corresponding modular form over $\C$ has a label of the form \(N.k.a.x.n.i\), where # - \(n\) determines the Conrey label \(N.n\) of the Dirichlet character \(\chi\); # - \(i\) is an integer ranging from 1 to the relative dimension of the newform that distinguishes embeddings with the same character $\chi$. # Dim -- # The **dimension** of a space of modular forms is its dimension as a complex vector space; for spaces of newforms $S_k^{\rm new}(N,\chi)$ this is the same as the dimension of the $\Q$-vector space spanned by its eigenforms. # The **dimension** of a newform refers to the dimension of its newform subspace, equivalently, the cardinality of its newform orbit. This is equal to the degree of its coefficient field (as an extension of $\Q$). # The **relative dimension** of $S_k^{\rm new}(N,\chi)$ is its dimension as a $\Q(\chi)$-vector space, where $\Q(\chi)$ is the field generated by the values of $\chi$, and similarly for newform subspaces. #$A$ (analytic_conductor) -- # The **analytic conductor** of a newform $f \in S_k^{\mathrm{new}}(N,\chi)$ is the positive real number # \[ # N\left(\frac{\exp(\psi(k/2))}{2\pi}\right)^2, # \] # where $\psi(x):=\Gamma'(x)/\Gamma(x)$ is the logarithmic derivative of the Gamma function. #Field (nf_label) -- # The **coefficient field** of a modular form is the subfield of $\C$ generated by the coefficients $a_n$ of its $q$-expansion $\sum a_nq^n$. The space of cusp forms $S_k^\mathrm{new}(N,\chi)$ has a basis of modular forms that are simultaneous eigenforms for all Hecke operators and with algebraic Fourier coefficients. For such eigenforms the coefficient field will be a number field, and Galois conjugate eigenforms will share the same coefficient field. Moreover, if $m$ is the smallest positive integer such that the values of the character $\chi$ are contained in the cyclotomic field $\Q(\zeta_m)$, the coefficient field will contain $\Q(\zeta_m)$ # For eigenforms, the coefficient field is also known as the **Hecke field**. #CM (cm_discs) -- # A newform $f$ admits a **self-twist** by a primitive # Dirichlet character $\chi$ if the equality # \[ # a_p(f) = \chi(p)a_p(f) # \] # holds for all but finitely many primes $p$. # For non-trivial $\chi$ this can hold only when $\chi$ has order $2$ and $a_p=0$ for all primes $p$ not dividing the level of $f$ for which $\chi(p)=-1$. # The character $\chi$ is then the Kronecker character of a quadratic field $K$ and may be identified by the discriminant $D$ of $K$. # If $D$ is negative, the modular form $f$ is said to have complex multiplication (CM) by $K$, and if $D$ is positive, $f$ is said to have real multiplication (RM) by $K$. The latter can occur only when $f$ is a modular form of weight $1$ whose projective image is dihedral. # It is possible for a modular form to have multiple non-trivial self twists; this occurs precisely when $f$ is a modular form of weight one whose projective image is isomorphic to $D_2:=C_2\times C_2$; in this case $f$ admits three non-trivial self twists, two of which are CM and one of which is RM. #RM (rm_discs) -- # A newform $f$ admits a **self-twist** by a primitive # Dirichlet character $\chi$ if the equality # \[ # a_p(f) = \chi(p)a_p(f) # \] # holds for all but finitely many primes $p$. # For non-trivial $\chi$ this can hold only when $\chi$ has order $2$ and $a_p=0$ for all primes $p$ not dividing the level of $f$ for which $\chi(p)=-1$. # The character $\chi$ is then the Kronecker character of a quadratic field $K$ and may be identified by the discriminant $D$ of $K$. # If $D$ is negative, the modular form $f$ is said to have complex multiplication (CM) by $K$, and if $D$ is positive, $f$ is said to have real multiplication (RM) by $K$. The latter can occur only when $f$ is a modular form of weight $1$ whose projective image is dihedral. # It is possible for a modular form to have multiple non-trivial self twists; this occurs precisely when $f$ is a modular form of weight one whose projective image is isomorphic to $D_2:=C_2\times C_2$; in this case $f$ admits three non-trivial self twists, two of which are CM and one of which is RM. #Traces (trace_display) -- # For a newform $f \in S_k^{\rm new}(\Gamma_1(N))$, its **trace form** $\mathrm{Tr}(f)$ is the sum of its distinct conjugates under $\mathrm{Aut}(\C)$ (equivalently, the sum under all embeddings of the coefficient field into $\C$). The trace form is a modular form $\mathrm{Tr}(f) \in S_k^{\rm new}(\Gamma_1(N))$ whose $q$-expansion has integral coefficients $a_n(\mathrm{Tr}(f)) \in \Z$. # The coefficient $a_1$ is equal to the dimension of the newform. # For $p$ prime, the coefficient $a_p$ is the trace of Frobenius in the direct sum of the $\ell$-adic Galois representations attached to the conjugates of $f$ (for any prime $\ell$). When $f$ has weight $k=2$, the coefficient $a_p(f)$ is the trace of Frobenius acting on the modular abelian variety associated to $f$. # For a newspace $S_k^{\rm new}(N,\chi)$, its trace form is the sum of the trace forms $\mathrm{Tr}(f)$ over all newforms $f\in S_k^{\rm new}(N,k)$; it is also a modular form in $S_k^{\rm new}(\Gamma_1(N))$. # The graphical plot displayed in the properties box on the home page of each newform or newspace is computed using the trace form. #Fricke sign (fricke_eigenval) -- # The **Fricke involution** is the Atkin-Lehner involution $w_N$ on the space $S_k(\Gamma_0(N))$ (induced by the corresponding involution on the modular curve $X_0(N)$). # For a newform $f \in S_k^{\textup{new}}(\Gamma_0(N))$, the sign of the functional equation satisfied by the L-function attached to $f$ is $i^{-k}$ times the eigenvalue of $\omega_N$ on $f$. So, for example when $k=2$, the signs swap, and the analytic rank of $f$ is even when $w_N f = -f$ and odd when $w_N f = +f$. #$q$-expansion (qexp_display) -- # The **$q$-expansion** of a modular form $f(z)$ is its Fourier expansion at the cusp $z=i\infty$, expressed as a power series $\sum_{n=0}^{\infty} a_n q^n$ in the variable $q=e^{2\pi iz}$. # For cusp forms, the constant coefficient $a_0$ of the $q$-expansion is zero. # For newforms, we have $a_1=1$ and the coefficients $a_n$ are algebraic integers in a number field $K \subseteq \C$. # Accordingly, we define the **$q$-expansion** of a newform orbit $[f]$ to be the $q$-expansion of any newform $f$ in the orbit, but with coefficients $a_n \in K$ (without an embedding into $\C$). Each embedding $K \hookrightarrow \C$ then gives rise to an embedded newform whose $q$-expansion has $a_n \in \C$, as above.